Introduction to Classical Mechanics · Problem 11.49

Relativity (Kinematics) — Morin

Physics — Mechanics · some-gammas

Problem

Show that the relativistic addition (or subtraction) of the velocities uu and vv has a γ\gamma factor given by γ=γuγv(1±uv)\gamma = \gamma_u \gamma_v (1 \pm uv).
Intuitive explanation

The key is recognizing that relativistic velocity addition mixes space and time in a specific way through Lorentz transformations. When you combine two velocity boosts, their γ\gamma factors don't simply multiply—instead, the relative motion between frames introduces a correction term involving the product uvuv (in units where c=1c=1). Work through how velocities transform between frames, and you'll see this γ\gamma factor emerge naturally from the structure of the Lorentz transformation itself.

Intuitive explanation

The key is recognizing that relativistic velocity addition mixes space and time in a specific way through Lorentz transformations. When you combine two velocity boosts, their γ\gamma factors don't simply multiply—instead, the relative motion between frames introduces a correction term involving the product uvuv (in units where c=1c=1). Work through how velocities transform between frames, and you'll see this γ\gamma factor emerge naturally from the structure of the Lorentz transformation itself.

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